if a and b are rational number, then find the value of a and b in 3+ √5= a+b√7.
3- √5
Answers
Step-by-step explanation:
What is the value of a and b, if (3+√5) / (3-√5) =a+b√5 where a and b are rational numbers?
(3 + sqrt(5)) / (3 - sqrt(5)) = (a + b*sqrt(5)) /1 , we have one equation of 2 fractions.
The cross product gives: (3 + sqrt(5)) = (3 - sqrt(5)) * (a + b*sqrt(5))
Getting rid of the brackets: 3 + sqrt(5) = 3a + 3b*sqrt(5) - 5b - a* sqrt(5)
3 + sqrt(5) = (3a -5b) + (3b - a)*sqrt(5) Separating the Rational from the Irrational,
we get 2 linear equations: 3 = (3a -5b) and
1 = (3b - a)
Multiplying the second equation by 3 gives: 3 = - 3a + 9b
adding the first equation: 3 = 3a -5b
we get : 3+3= - 3a + 3a + 9b - 5b
Eliminating the variable a : 6 = 4 b this gives b = 6/4 = 3/2
Now knowing b=3/2 , we substitute it in 1 = (3b - a) and we get a = 7/2
The answer is: a = 7/2 ; and b = 3/2